It costs 2 Doom to acquire the Loyalty card for the Prophetess or Eidolon and bring them onto the map. When is this done: at any time, during the Doom Phase, during the Action Phase, or?

Once in play they always belong to the player who paid the Doom and holds the Loyalty card, but if sent back to his pool by elimination of some sort they can be Recruited by any player. If eliminated again they return to the “Loyalty” player’s pool, and are again available for Recruitment by anyone who performs the Recruit Action and pays the Power cost to do so, and function as a Cultist for the Recruiting faction. Have I got that right?